Volatility Skew

Consider put or call options on a given underlier. They have different strikes but the same expiration. If we obtain market prices for those options, we can apply the Black-Scholes (1973) model to back-out implied volatilities. Intuitively, we might expect the implied volatilities to be identical. In practice, it is likely that they will not be.

Coffee options trade on New York’s Coffee, Sugar and Cocoa Exchange (CSCE).

The pattern of implied volatilities forms a “smile” shape, which is called a volatility smile. Such a smile persists over time in the coffee options markets with in-the-money and out-of-the-money volatilities generally higher than at-the-money volatilities.

Most derivatives markets exhibit persistent patterns of volatilities varying by strike. In some markets, those patterns form a smile. In others, such as equity index options markets, it is more of a skewed curve. This has motivated the name volatility skew. In practice, either the term “volatility smile” or “volatility skew” (or simply skew) may be used to refer to the general phenomena of volatilities varying by strike. Indeed, you may even hear of “volatility smirks” or “volatility sneers”, but such names are often as much whimsical as they are descriptive of any particular volatility pattern.

There are various explanations for why volatilities exhibit skew. Different explanations may apply in different markets. In most cases, multiple explanations may play a role. Some explanations relate to the idealized assumptions of the Black-Scholes approach to valuing options. Almost every one of those assumptions—lognormally distributed returns, return homoskedasticity, etc.—could play a role. For example, in most markets, returns appear more leptokurtic than is assumed by a lognormal distribution. Market leptokurtosis would make way out-of-the-money or way in-the-money options more expensive than would be assumed by the Black-Scholes formulation. By increasing prices for such options, volatility smile could be the markets’ indirect way of achieving such higher prices within the imperfect framework of the Black-Scholes model. Other explanations relate to relative supply and demand for options. In equity markets, volatility skew could reflect investors’ fear of market crashes—which would cause them to bid up the prices of options at strikes below current market levels. In electricity markets, utilities and other purchasers are concerned about price spikes. Not surprisingly, electricity volatilities exhibit the opposite skew—with volatilities elevated for higher strikes.

Another dimension to the problem of volatility skew is that of volatilities varying by expiration.

For early expirations, the graph exhibits a volatility smile. Volatilities dip and then rise, taking on a skew for the September contract before falling and flattening out. This is a pattern that recurs every year for coffee volatilities. September is the harvest month for Brazilian coffees. Brazil is a major exporter, and every year, the harvest is threatened by frost. Market participants watch the harvest, knowing that a frost will drive world coffee prices sharply higher. This explains both the rise in volatilities as well as the skew for September. For that month, traders are concerned about a spike in prices, and the volatility surface reflects this.

If we take a cross-section of a volatility surface at a particular strike, we obtain a curve that describes implied volatilities as a function of expiration for that strike. This is called a volatility term structure.

Traders aren’t interested only in static volatility surfaces. They also want to know how skew will respond to the passage of time and changes in the underlier’s value. While the dynamics of a volatility surface are complicated, there are two simple models that are useful for describing an aspect of those dynamics. Introduced by Derman (1999), these are called the sticky strike and sticky delta models.

If a skew is behaving according to the sticky strike model, its implied volatilities are associated with specific strikes. The curve of the skew will not shift with the value of the underlier.

With the sticky delta model, implied volatilities are associated with specific deltas. The entire curve of volatilities moves with the underlier. For example, suppose a strike 75 call has a delta of .65 and implied volatility of 14%. If the value of the underlier rises so that the strike 80 option becomes the delta .65 option, then the 14% implied volatility will migrate to that option.

Obviously, both models are simplifications. No market exhibits one behaviour or the other at all times, and actual volatility dynamics often blend the two.

Volatility skew complicates the tasks of pricing and hedging options. Consider the task of calculating an option’s delta. If we assume the sticky delta model, this will affect how we calculate the option’s delta. Changes in implied volatilities that are expected to accompany changes in the value of the underlier will impact the option’s value. Deltas need to be adjusted to reflect this. This is more than a theoretical consideration. If a trader is dynamically hedging an options position and fails to incorporate skew into her delta calculations, her hedge ratio will be off.

Skew poses even greater challenges for financial engineers. They need to adopt dynamic models for entire volatility surfaces. These are far more sophisticated than the simple sticky strike and sticky delta models. Standard models include

• Merton’s (1976) jump-diffusion model;
• stochastic volatility models of Hull and White (1988) and Heston (1993);
• local volatility models of Dupire (1994), Derman and Kani (1994), Rubinstein (1994) and Andersen and Brotherton-Ratcliffe (1997);
• mixed distribution models such as Brigo and Mercurio (2000).

A jump-diffusion model adds random jumps to the geometric Brownian motion that Black-Scholes (1973) assumes for the underlier. Among other things, this has the effect of giving the underlier’s value a leptokurtic distribution.

A stochastic volatility model models both the underlier’s value and its volatility as stochastic processes. As with a jump-diffusion model, this has the effect of giving the underlier’s value a leptokurtic distribution. Heston’s (1993) is a popular stochastic volatility model.

Local volatility models have various names. They have been called deterministic volatility function models or implied tree models (implied binomial tree or implied trinomial tree models, depending on the type of tree). They assume that future volatilities will be a deterministic function of the underlier value and time. This function is implied by the current volatility skew and can be fully reflected in a suitably calibrated binomial or trinomial tree.

Mixed distribution models model the underlier’s value with a mixture of distributions. This has the effect of giving the underlier a leptokurtic distribution. Brigo and Mercurio (2000) use a mixture of lognormals.

Other models are hybrids, combining aspects of those described above. Bates’ (1996) stochastic volatility jump-diffusion model is an example. The universal volatility models of Dupire (1996) and Britten-Jones and Neuberger (2000), among others, combine elements of local volatility, jump-diffusion and stochastic volatility models. These are rich models, but they can be difficult to implement.

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